<html>
  <head>
    <meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
    <title>feedback</title>
  </head>
  <body bgcolor="#FFFFFF">
    <center>Scilab Function</center>
    <div align="right">Last update : April 1993</div>
    <p>
      <b>feedback</b> -  feedback operation</p>
    <h3>
      <font color="blue">Calling Sequence</font>
    </h3>
    <dl>
      <dd>
        <tt>Sl=Sl1/.Sl2</tt>
      </dd>
    </dl>
    <h3>
      <font color="blue">Parameters</font>
    </h3>
    <ul>
      <li>
        <tt>
          <b>Sl1,Sl2</b>
        </tt>: linear systems (<tt>
          <b>syslin</b>
        </tt> list) in state-space or transfer form, or ordinary gain matrices.</li>
      <li>
        <tt>
          <b>Sl</b>
        </tt>: linear system (<tt>
          <b>syslin</b>
        </tt> list) in state-space or transfer form</li>
    </ul>
    <h3>
      <font color="blue">Description</font>
    </h3>
    <p>
    The feedback operation is denoted by <tt>
        <b> /. </b>
      </tt> (slashdot).
    This command returns <tt>
        <b>Sl=Sl1*(I+Sl2*Sl1)^-1</b>
      </tt>, i.e the (negative) 
    feedback of <tt>
        <b>Sl1</b>
      </tt> and <tt>
        <b>Sl2</b>
      </tt>. <tt>
        <b>Sl</b>
      </tt> is the transfer
    <tt>
        <b> v -&gt; y</b>
      </tt> for <tt>
        <b> y = Sl1 u</b>
      </tt>, <tt>
        <b>u = v - Sl2 y</b>
      </tt>.</p>
    <p>
    The result is the same as <tt>
        <b>Sl=LFT([0,I;I,-Sl2],Sl1)</b>
      </tt>.</p>
    <p>
    Caution: do not use with decimal point (e.g. <tt>
        <b>1/.1</b>
      </tt> is ambiguous!)</p>
    <h3>
      <font color="blue">Examples</font>
    </h3>
    <pre>

S1=ssrand(2,2,3);S2=ssrand(2,2,2);
W=S1/.S2;
ss2tf(S1/.S2)
//Same operation by LFT:
ss2tf(lft([zeros(2,2),eye(2,2);eye(2,2),-S2],S1))
//Other approach: with constant feedback
BigS=sysdiag(S1,S2); F=[zeros(2,2),eye(2,2);-eye(2,2),zeros(2,2)];
Bigclosed=BigS/.F;
W1=Bigclosed(1:2,1:2);   //W1=W (in state-space).
ss2tf(W1)
//Inverting
ss2tf(S1*inv(eye()+S2*S1))
 
  </pre>
    <h3>
      <font color="blue">See Also</font>
    </h3>
    <p>
      <a href="../robust/lft.htm">
        <tt>
          <b>lft</b>
        </tt>
      </a>,&nbsp;&nbsp;<a href="../elementary/sysdiag.htm">
        <tt>
          <b>sysdiag</b>
        </tt>
      </a>,&nbsp;&nbsp;<a href="../robust/augment.htm">
        <tt>
          <b>augment</b>
        </tt>
      </a>,&nbsp;&nbsp;<a href="obscont.htm">
        <tt>
          <b>obscont</b>
        </tt>
      </a>,&nbsp;&nbsp;</p>
  </body>
</html>
